The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product

We consider the Belkale-Kumar cup product $\odot_t$ on $H^*(G/P)$ for a generalized flag variety $G/P$ with parameter $t \in \C^m$, where $m=\dim(H^2(G/P))$. For each $t\in \C^m$, we define an associated parabolic subgroup $P_K \supset P$. We show that the ring $(H^*(G/P), \odot_t)$ contains a graded subalgebra $A$ isomorphic to $H^*(P_K/P)$ with the usual cup product, where $P_K$ is a parabolic subgroup associated to the parameter $t$. Further, we prove that $(H^*(G/P_K), \odot_0)$ is the quotient of the ring $(H^*(G/P), \odot_t)$ with respect to the ideal generated by elements of positive degree of $A$. We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.


Introduction
Let G be a complex semisimple algebraic group with parabolic subgroup P of G. Let m = dim(H 2 (G/P )). For each t ∈ C m , Belkale and Kumar defined a product ⊙ t which degenerates the usual cup product on H * (G/P ), and gave striking applications of this product to the eigenvalue problem and to the problem of finding G-invariants in tensor products of representations [BK]. In [EG], we gave a new construction of this product, and showed that the ring (H * (G/P, C), ⊙ t ) is isomorphic to a relative Lie algebra cohomology ring H * (g t , l ∆ ); here g t ⊃ l ∆ are certain subalgebras of g × g.
In this paper, we study the ring H * (G/P, ⊙ t ). Let α 1 , . . . , α n be the simple roots with respect to a Borel subgroup B ⊂ P and a Cartan subgroup H ⊂ B. Let L be the Levi factor of P containing H, let l be the Lie algebra of L, and number the simple roots so that I = {α m+1 , . . . , α n } are roots of l, and α 1 , . . . , α m are roots of u, the nilradical of the Lie algebra of P . For each t = (t 1 , . . . , t m ) ∈ C m , let J(t) = {1 ≤ q ≤ m : t q = 0}, and let K = J(t) ∪ I. Let l K be the Levi subalgebra generated by the Lie algebra h of H and the root spaces g ±α i for i ∈ K, and let L K be the corresponding subgroup. Let P K = BL K be the corresponding standard parabolic.
Theorem 1.1. For parabolic subgroups P ⊂ P K of G, with P K determined by t ∈ C m as above, (1) H * (P K /P ) is isomorphic to a graded subalgebra A of (H * (G/P ), ⊙ t ).
Evens was supported by the National Security Agency. Mathematics Subject Classification: 17B56, 14M15, 20G05.
This theorem asserts in effect that (H * (G/P ), ⊙ t ) has a classical part isomorphic to the usual cohomology ring, with associated quotient given by the degenerate Belkale-Kumar product. Using the relative Lie algebra cohomology description of the product, this theorem follows from the Hochschild-Serre spectral sequence in relative Lie algebra cohomology. To make this argument rigorous, one needs to know certain facts about the spectral sequence. In particular, one must show that the spectral sequence degenerates at the E 2 -term, compute the edge morphisms, and determine the product structure on the E 2 -term. The main point of this paper is to carry this out using the original approach of Hochschild and Serre [HS]. While Borel and Wallach give an excellent treatment of representation theoretic aspects of the Hochschild-Serre spectral sequence in [BW], they do not discuss the ring structure, and we are aware of no general reference for the facts needed to prove our theorem. The Hochschild-Serre spectral sequence is of general utility, and one of the goals of this paper is to provide a careful treatment of these facts for the literature. Also, the construction of the relative Hochschild-Serre spectral sequence in [BW] uses the identification of the relative Lie algebra cohomology groups with derived functors. In [EG] the definition of these groups using cochains was crucial in order to be able to apply results of Kostant from [Kos2]. By proving what we need using the cochain definition, we avoid the necessity of defining the ring structure in the Borel-Wallach setting, and of proving the compatibility of the two settings.
The approach of Hochschild and Serre generalizes in a mostly straightforward fashion to the relative setting, but there is one new point. In this setting we have a Lie algebra g, an ideal I and a subalgebra k which is reductive in g. To construct the spectral sequence we need an action of g/I on the relative cohomology group H * (I, I ∩ k; M ) (here M is a g-module). If I ∩ k is nonzero, the Lie algebra g/I does not act in an obvious way on the space of cochains C * (I, I ∩ k; M ). Nevertheless, we are able to define the action on the cohomology group by a formula involving cochains; verifying that this does yield the d 1 differential in the spectral sequence is the main technical complication of the paper. In fact, in the Belkale-Kumar application, I ∩ k = 0, so this complication can be avoided, but it seemed worthwhile to develop the spectral sequence without this potentially limiting assumption.
The contents of the paper are as follows. In Section 2 we give basic facts about a Lie algebra g equipped with I and k. We also recall some basic definitions related to Lie algebra cohomology, and define the action of g/I on H * (I, I ∩ k; M ). In Section 3 we study the filtration on cochains introduced by Hochschild and Serre and extend their results to the relative setting. In Section 4, we prove some formulas involving differentials which are used in calculating the differentials in the spectral sequence. Section 5 recalls basic definitions and facts about spectral sequences. Section 6 proves the existence of the Hochschild-Serre spectral sequence in the relative setting, identifies the edge maps and proves some results on the product structure. Finally, Section 7 applies these results to the Belkale-Kumar product.

Preliminaries
We work over a field F of arbitrary characteristic. In Section 7, we take F = C. Let g be a Lie algebra, I ⊂ g an ideal, and k ⊂ g a subalgebra. Write I k = I ∩ k. We assume that k is reductive in g. This is equivalent to the statement that if U ⊂ V are k-submodules of g, then U has a k-invariant complement U ′ , so V = U ⊕ U ′ .
Lemma 2.1. There is a k-module decomposition of g given by Proof. Let I L be a k-module complement of I k in I, and let J k be a k-module complement of I k in k. Then Let J L be a k-module complement to I + k in g. The decomposition satisfies (2.2). Moreover, [J, I k ] ⊂ J ∩ I = 0, so J and I k commute.
We do not assert that J is a subalgebra of g, only a k-submodule. We have a kmodule decomposition g = I ⊕ J. Let π : g → I denote the projection arising from this decomposition. We will sometimes write x * for π(x). Let x → x + = x − x * denote the projection g → J.
The proof is immediate.
2.1. Lie algebra cohomology. In this section we recall some of the basic definitions of Lie algebra cohomology. Let M be a g-module, and let C n (g; M ) = Hom(Λ n g, M ).
We identify this space with the space of alternating n-linear maps from g × · · · × g to M . Given x = (x 1 , . . . , x n ) ∈ g n and z ∈ g, write The Lie algebra cohomology differential d : We remark that for any vector space V and any subspace W ⊂ V we always view C n (g/I; V ) = Hom F (Λ n (g/I), V ) as a subspace of C n (g; V ), so as arguments to f ∈ C n (g/I; V ), we can allow elements of g or of g/I. We adopt the analogous viewpoint for other spaces.
Define θ z : C n (g; M ) → C n (g; M ) and i x : We will often write z · f for θ z f . Observe that since I is an ideal of g, the same formula defines θ z : C n (I; M ) → C n (I; M ).
Lemma 2.7. The differentials on C(g; M ) and on C(I; M ) commute with the g-action.
Proof. The pullback γ * : C(g; M ) → C(I; M ) is surjective and commutes with d and with the g action. Therefore it suffices to check that the g-action on C(g; M ) commutes with d. The element z ∈ g acts by θ z , and the fact that θ z commutes with d is an immediate consequence of (2.6).
Let C n (g, k; M ) denote the subspace of C n (g; M ) consisting of those elements annihilated by i x and θ x for all x ∈ k, i.e., the subset of linear maps f : for all x ∈ k. In other words, we can identify It is well-known that C(g, k; M ) is a subcomplex of C(g; M ).
Let Z q (g, k; M ) and B q (g, k; M ) denote the groups of cocycles and coboundaries in C q (g, k; M ). The q-th cohomology group of the complex C(g, k; M ) is denoted H q (g, k; M ). In the absolute case, Z q (g; M ) and B(g; M ) denote the groups of cocycles and coboundaries in C q (g; M ), and H q (g; M ) the cohomology. We adopt analogous notation for other Lie algebras.
Pairings on relative Lie algebra cohomology are discussed in Section 6.4.
2.2. The g/I-action on H * (I, I k ; M ). In this section we define an action of g/I on H q (I, I k ; M ) using the complex C(I, I k ; M ) (Borel and Wallach define a g/I-action using injective resolutions).
Lemma 2.8. If x ∈ J, then the operator θ x on C q (I; M ) preserves the subspace C q (I, I k ; M ).
Proof. Let c ∈ C q (I, I k ; M ). We must check that for y ∈ I k , we have θ y θ x c = 0 and i y θ x c = 0. The first equation holds because J and I k commute (Lemma 2.1); thus, θ y and θ x commute, so θ y θ x c = θ x θ y c = 0, as θ y c = 0. To check the second equation, let y 1 ∈ I k , and y 2 , . . . , y q ∈ I. Then The term c([y 1 , x], y 2 , . . . , y q ) is zero because [y 1 , x] = 0. All other terms have y 1 as an argument of c; as i y 1 c = 0, these terms give 0.
If x ∈ J, then θ x commutes with the differential d on C q (I; M ) (by Lemma 2.7) and preserves the subspace C q (I, I k ; M ) of C q (I; M ). Hence θ x preserves the subspaces Z q (I, I k ; M ) and B q (I, I k ; M ), so there is an induced action of θ x on the cohomology H q (I, I k ; M ).
As noted previously, we will often write simply x · c for θ x c. Recall the projection g → J, x → x + . Proposition 2.9. There is a Lie algebra action of g on H q (I, I k ; M ), defined by the formula x · [c] := [x + · c] (where x + · c means θ x + c). Since I acts trivially, this formula defines an action of g/I on H q (I, I k ; M ).
Proof. We must check that the above definition is compatible with the Lie bracket, i.e., that for x, y ∈ g, c ∈ Z q (I, I k ; M ), we have (2.10) The Next, observe that u commutes with I k , as by Lemma 2.1, x + , y + , and [x, y] + all commute with I k . We conclude that u ∈ Z I (I k ).
Thus, u · c ∈ B q (I; M ), and to show that u · c ∈ B q (I, I k : M ), we must show that the element i u c of C q−1 (I; M ) lies in the subspace C q−1 (I, I k ; M ). In other words, we must show that for a ∈ I k , we have i a i u c = 0 and θ a i u c = 0. The first equation holds because i a i u c = −i u i a c = 0 as i a c = 0. To verify the second equation, observe that where in the second equality, we have used i a i u c = 0, and in the third equality, we have used dc = 0. If y 2 , . . . , y q ∈ I, then (i a θ u c)(y 2 , . . . , y q ) = (θ u c)(a, y 2 , . . . , y q ) = u(c(a, y 2 , . . . , y q )) + c([a, u], y 2 , . . . , y q ) + q i=2 c(a, y 2 , . . . , [y i , a], . . . , y q )). This is 0 because i a c = 0 (so we get 0 if any input to c is equal to a) and [a, u] = 0. Hence θ a i u c = 0. We conclude that i u c ∈ C q−1 (I, I k ; M ), so θ u c = di u c ∈ B q (I, I k ; M ), completing the proof. Proof. Let c ∈ C n (g, k; M ) be a cocycle. We must show that j * [c] is g/I-invariant, i.e. if x ∈ g, then x · (j * [c]) = 0. By definition, So we want to show that θ x + j * c is the coboundary of an element in C n−1 (I, I k ; M ). We have provided the equation is interpreted correctly: c ∈ C n (g, k; M ) ⊂ C n (g; M ), but since we do not know that C n (g, k; M ) is stable under θ x + , we must view θ x + c as an element of C n (g; M ). We have We know that j * i x + c is an element of C n−1 (I; M ). To complete the proof, we must show that it lies in the subspace C n−1 (I, I k ; M ). Let y ∈ I k . Then i y j * i x + c = −j * i x + i y c, and this is zero since c ∈ C n (g, k; M ). Also, given x 2 , . . . , x n ∈ I, we have (θ y j * i x + c)(x 2 , . . . , x n ) = y · (c(x + , x 2 , . . . , x n )) + i≥2 c(x + , x 2 , . . . , [x i , y], . . . , x n ).
(2.13) Lemma 2.1 implies that [x + , y] = 0, so the right side of (2.13) is unchanged if we add the term c([x + , y], x 2 , . . . , x n ). But if we add this term, the right hand side of (2.13) is by definition equal to (θ y c)(x + , x 2 , . . . , x n ). But θ y c = 0 since c ∈ C n (g, k; M ). We conclude that j * i x + c is in C n−1 (I; M ), completing the proof.

The filtration and consequences
For simplicity, except for Section 5, we will now write C n = C n (g, k; M ). This space has a filtration (introduced by Hochschild and Serre in the absolute setting): by definition, F p C n (g; M ) consists of the subspace of C n (g; M ) consisting of those f which are zero when n − p + 1 of the inputs are in I. In other words, f (x 1 , . . . , x n ) can be nonzero only if at most n − p of the x 1 , . . . , x n are in I. Given a subspace . In particular, this defines F p C n . As observed by Hochschild and Serre, the Lie algebra cohomology differential d takes This is the 0-th page of the Hochschild-Serre spectral sequence (we recall spectral sequence generalities in Section 5). Our immediate goal is to describe E pq 0 . Since k/I k acts on g/I, we can define the vector space Note that g/I k/I k ∼ = g/(I + k), so C p (g/I, k/I k ; C q (I, I k ; M )) = Hom k/I k (Λ p (g/(I + k)), C q (I, I k ; M )).
, and s p the restriction of S p to F p C p+q .
Proof. We must show that if f is k-invariant, then so is S p f . Let z ∈ k, x 1 , . . . , x p ∈ g, and y 1 , . . . , y q ∈ I. Now, k-equivariance means that if z ∈ k and x ∈ g p , then To verify this equation, we must evaluate both sides at y ∈ g q . The left side gives while the right side gives f (z · x, y). (3.6) The expressions (3.5) and (3.6) are equal because f is k-equivariant. Hence S p f is k-equivariant, as desired.
We clarify some notation regarding interior products. Note that if f ∈ C p (g, C q (g; M )) and y ∈ g, then i y f ∈ C p−1 (g, C q (g; M )). Given a p-tuple . We will use the same notation for maps restricted to subspaces of C p+q (g; M ) and C p (g; C q (g; M )).
The following lemma generalizes to the relative situation a result from [HS]; the arguments are adapted from there. The lemma implies that s p induces an isomorphism ). This is our desired description of E pq 0 . Lemma 3.7. ( Proof. (1) The statement about the kernel of s p is immediate from the definition of the filtration.
(2) Suppose f ∈ F p C p+q . We want to show that s p (f ) ∈ C p (g/I, k/I k ; C q (I, I k ; M )). Since s p is the restriction of S p and of r p , we know that s p f lies in the intersection of the images of S p and of r p , i.e., in C p (g, k; C q (I; M )) ∩ C p (g/I; C q (I; M )).
To prove (2), it will be enough to show that for x ∈ g p , we have i x f ∈ C q (I, I k ; M ), for then the notions of k-equivariance and k/I k -equivariance will coincide.
To show that i x f ∈ C q (I, I k ; M ), we must verify that for z ∈ I k , we have The first equation holds because if an element of k is an argument of f , the result is 0. For the second equation, since f ∈ C p+q (g, k; M ), and z ∈ I k ⊂ k, we have θ z f = 0. By definition of θ z , this means that for x ∈ g p and y ∈ I q , we have The first term on the right is 0 because it is a sum of terms, each of which has at least q + 1 inputs from I. Thus, we obtain As y is arbitrary, we see that θ z i x s p f = 0, as desired.
(3) We now prove that s p is surjective. We must show that if g ∈ C p (g/I, k/I k ; C q (I, I k ; M )), then there exists f ∈ F p C p+q such that s p f = g.
As in [HS] Here, the map x → x * is as in Corollary 2.3. The sum is over all permutations σ ∈ Σ p+q such that σ(1) < σ(2) < · · · < σ(p) and σ(p + 1) < σ(p + 2) < · · · < σ(p + q). (We have asserted that f ∈ C p+q (g; M ); this amounts to checking that f is alternating-see Remark 3.11 below.) We must verify that f ∈ C p+q (g, k; M ), i.e., that if x ∈ k, then i x f = 0 and θ x f = 0. To show the first equation, it suffices to show that if x 1 ∈ k, then f (x 1 , . . . , x p+q ) = 0. This holds because either σ(1) = 1, in which case we get 0 as i x 1 g = 0; or σ(p + 1) = 1, in which case x * 1 ∈ I k by Corollary 2.3, so we get 0 because i x g ∈ C q (I, I k ; M ). Now we show that θ x f = 0. This says that for z ∈ k and x 1 , . . . , x p+q ∈ g, we have (3.9) By assumption, g is k/I k -equivariant (i.e., k-invariant, and I k acts trivially). Thus, for z ∈ k and x 1 , . . . , x p ∈ g, we have (3.10) Given (x p+1 , . . . , x p+q ) ∈ g q , we have (x * p+1 , . . . , x * p+q ) ∈ I q , to which we can apply both sides of the preceding equation. Upon rearranging, by definition of the k-action on C q (I, I k ; M ), we obtain This is exactly the equation we obtain by taking the equation (3.9) of k-equivariance of f and expressing f in terms of g. This proves (3.9), so f is k-equivariant. We conclude that f ∈ C p+q (g, k; M ), as desired.
Next, observe that f ∈ F p C p+q (g, k; M ). The reason is that if more than q of the arguments of f are in I, then for any permutation σ, at least one of the x σ(1) , . . . , x σ(p) must be in I, in which case g(x σ(1) , . . . , x σ(p) )(x * σ(p+1) , . . . , x * σ(p+q) ) = 0. Finally, we must verify that s p f = g. For this, suppose that x 1 , . . . , x p are in g and x p+1 , . . . , x p+q are in I. Then in the definition of f , the only permutation that contributes is the identity permutation, and we see that . , x * σ(p+q) ) = 0 by the alternating properties of g. Otherwise, the contribution to (3.8) from σ is the negative of the contribution from τ σ, where τ is the transposition (i i + 1). We conclude that f (x 1 , . . . , x p+q ) = 0, as desired.
Definition 3.12. For later use, we will denote by g → g the map C p (g/I, k/I k ; C q (I, I k ; M )) → F p C p+q constructed in the preceding proof; that is, g is the element f defined in (3.8). The preceding proof shows that s p g = S p ( g) = g.

Some formulas involving differentials
We give here some facts involving various Lie algebra cohomology differentials, which will be used in Section 6 when we calculate the differentials in the spectral sequence.
where the d on the right hand side of the equation is the differential C q (I; M ) → C q+1 (I; M ). Second, there is the horizontal differential d h : C p,q → C p+1,q (this is the Lie algebra cohomology differential d : C p (g; C q (I; M )) → C p+1 (g; C q (I; M )), with C q (I; M ) playing the role of the module; note that x ∈ g acts on C q (I; M ) using the action θ x defined above).
Lemma 4.1. The differentials d h and d v (on C(g, C(I; M ))) commute.
Proof. Any g-module map V → W induces a map of complexes C(g; V ) → C(g; W ) which is compatible with the Lie algebra cohomology differential. By Lemma 2.7, the differential C q (I; M ) → C q+1 (I; M ) is a g-module map. The induced map C(g; C q (I; M )) → C(g; C q+1 (I; M )) is what we have denoted d v ; the compatibility with the Lie algebra cohomology differential (with g playing the role of Lie algebra) amounts to the commutativity of d h and d v .
Lemma 4.2. The space C(g, k; C(I, M )) (viewed as a subspace of C(g, C(I, M ))) is stable under d h and d v . Moreover, the spaces Proof. The space C(g, k; C(I, M )) is stable under d h because it is a relative Lie algebra cohomology complex (with C(I; M ) playing the role of module). To verify stability under d v , let f ∈ C p (g, k; C q (I; M )), and z ∈ k. We must show two things. First, for Indeed, suppose z ∈ k and x ∈ g p . Writing the action of z on C(I; M ) as θ z , we want to show that ). Since f is k-equivariant, this equals d(θ z (f (x))). By Lemma 2.7, d commutes with θ z . Hence C(g, k; C(I; M )) is stable under d v . The remaining assertions follow easily.
The following lemma is essentially given in [HS], so we omit the proof.
Lemma 4.3. Let f ∈ C p+q (g; M ). Then The analogous formula holds with R replaced by S.
4.1. Differentials related to relative cohomology. To streamline the exposition we introduce some notation.
Lemma 4.6. There is a map Proof. It is evident that d + z ∈ C p+1 (g/I; C q (I, I k ; M )). We must verify that d + z is k/I k -equivariant, or equivalently, is k-equivariant. We have C p (g/I, k/I k ; C q (I, I k ; M )) ⊂ C p (g, k; C q (I; M )) d h → C p+1 (g, k; C q (I; M )).
Therefore, changing j to i in this sum, we can rewrite (4.11) as We want to show that this is equal to (4.9), i.e., that This equality holds because the k-equivariance of z implies that u · z(y(i)) = z(u · y(i)).
Although d h commutes with d v , we do not assert that d + commutes with d v .
Notice that the formula for d + would be a Lie algebra cohomology differential if g/I acted on C q (I, I k ; M ), with the action of x taking c to x + · c. However, this formula does not define a Lie algebra action (i.e., it is not compatible with the Lie bracket). Thus, we cannot assert that d + • d + is zero on cochains. However, by Proposition 2.9, g/I does act on H q (I, I k ; M ) by the analogous formula, i.e., by x · [c] = [x + · c]. Write [z] for the image of z under the map C p (g/I, k/I k ; Z q (I, I k ; M )) → C p (g/I, k/I k ; H q (I, I k ; M )).
The Lie algebra cohomology differential C p (g/I, k/I k ; H q (I, I k ; M )) → C p+1 (g/I, k/I k ; H q (I, I k ; M )) is given by [z] → [d + z]. By abuse of notation we will simply write d + [z] = [d + z].
Lemma 4.12. Let f ∈ Z q (I; M ), and let x 0 , . . . , x q ∈ I. Then Proof. This follows by writing out the equation df (x 0 , . . . , x q ) = 0 and rearranging, using the definition of x 0 · f .
The following proposition and corollary relate z = S p z and S p+1 z.
(4.16) On the other hand, by Proposition 4.13, ( 4.17) Now, y i − y * i = y + i . The corollary follows by adding (4.16) and (4.17) and comparing the result with the definition of d + z.

The spectral sequence
5.1. Spectral sequence generalities. In this section we recall some standard facts about spectral sequences. Our basic source is Chapter 5 of [Wei], but we have modified some of the definitions for convenience. See also Chapter XX.9 of [Lan].
Suppose C = ⊕ n≥0 C n is a graded cochain complex with differential d. We assume also that C has a decreasing filtration (compatible with d): that is, for each n, we have F 0 C n ⊃ F 1 C n ⊃ · · · ; the filtration is extended to negative indices by setting F p C n = F 0 C n for p < 0. We also assume that for each n there exists some r (depending on n) such that F r C n = 0. Let B n ⊂ Z n denote the spaces of coboundaries and cocycles (respectively) in C n . These spaces are filtered by setting F p B n := F p C n ∩ B n and F p Z n := F p C n ∩ Z n . The cohomology H n (C) is filtered by F p H n (C) = (F p Z n + B n )/B n .

Define
E pq 0 = gr p C p+q := F p C p+q /F p+1 C p+q , and let π p : F p C p+q → E pq 0 denote the projection. For each r, n, p ≥ 0 let Thus, F p C n = F p C n (0) ⊃ F p C n (1) ⊃ . . .. We define subspaces of F p C p+q : first, Z pq r := F p C p+q (r) + F p+1 C p+q ⊃ B pq r := dF p−r+1 C p+q−1 (r − 1) + F p+1 C p+q .
Our assumption on the filtration means that for r sufficiently large, Z pq r = Z pq ∞ and There are inclusions of these spaces: The r-th page of the spectral sequence is defined to be An equivalent definition of the r-th page is sometimes more convenient. Given sub- . Using this second description, we define a differential d r : E pq r → E p+r,q−r+1 r by d r (c + G pq r ) = dc + G p+r,q−r+1 r .
The following proposition describes one of the key properties of spectral sequences.
Proposition 5.4. Under the surjection π p : Z pq r → E pq r , the inverse image of ker d r (resp. im d r ) is Z pq r+1 (resp. B pq r+1 ). Hence π p induces an isomorphism We omit the proof.
Remark 5.5. For r = 0 and r = 1, the definitions (5.1) and (5.3) of the r-th page are identical. In particular, G pq r = B pq r, and d r (z + B pq r ) = dz + B pq r . However, the analogous assertion is false for r > 1.
The spectral sequence is said to degenerate at E r if for all p, q, we have Z pq r = Z pq ∞ and B pq r = B pq ∞ . This is equivalent to the vanishing of all differentials d s , s ≥ r, and implies that E pq r = E pq ∞ . Observe that induces an isomorphism gr p H p+q (C) → E pq ∞ . Suppose that C has a graded algebra structure such that the differential d satisfies d(cc ′ ) = (dc)c ′ + (−1) p c(dc ′ ) for c ∈ C p , c ′ ∈ C. Then H(C) is a graded algebra, and gr H(C) is a bigraded algebra. Moreover, each E r has the structure of a bigraded algebra, and the map gr H(C) → E ∞ is an isomorphism of bigraded algebras. 5.2. Edge maps. Suppose that F p+1 C p = 0 for all p. Suppose also that r ≥ 2; in this case F p C p (r) = F p Z p . Consider the map The first map E p0 r → E p0 ∞ is surjective. If the spectral sequence degenerates at E r , we obtain E p0 r = gr p H p (C) ֒→ H p (C).
Next, making use of the facts that if r > 0 then F 1−r C q = C q , we have an edge map .
(5.7) Here the first map is surjective. If the spectral sequence degenerates at E r , we obtain a surjection H q (C) → gr 0 H q (C) = E 0q r .

The product structure. Let
We endow the tensor product A ⊗ B with an algebra structure such that (a 1 ⊗ b 1 ) · (a 2 ⊗ b 2 ) = (−1) q 1 p 2 a 1 a 2 ⊗ b 1 b 2 , The edge maps give an inclusion A ֒→ H(C), A p = gr p H p (C) ⊂ H p (C). Similarly, the edge maps give a surjection H(C) → B, H q (C) → gr 0 H q (C) = B q . Let J denote the kernel of the map H(C) → B; then J = ⊕ q F 1 H q (C).
Proposition 5.8. Suppose that the multiplication map A ⊗ B → E ∞ is an algebra isomorphism. Then: (1) The ideal J is equal to the ideal of H(C) generated by A + .
(2) If b 1 , . . . b n are homogeneous elements of H(C) whose images under the map H(C) → B form a basis of B, then these elements form an A-module basis of H(C). Hence if dim B = n is finite, then H(C) is a free A-module of rank n.

Proof.
(1) Let J ′ denote the ideal A + H(C). We want to show that J ′ = J. Since A + ⊂ F 1 H(C), we have J ′ ⊂ J. For the reverse inclusion, it suffices to show that if h ∈ F p H q (C) with p ≥ 1, then h ∈ J ′ . We use downward induction on p. If p > q then F p H q (C) = 0 so the result holds. Now suppose the result holds for elements of F p+1 H q (C). Let h denote the image of h in E p,q−p ∞ . By hypothesis we can write h = a q β q for some elements a q ∈ A p , β q ∈ B q−p . Choose elements b q ∈ H q−p (C) mapping to β q ∈ B. Then h− a q b q ∈ F p+1 H q (C) is in J ′ by the inductive hypothesis. Since a q b q ∈ J ′ , we conclude h ∈ J ′ , as desired.
(2) Let b 1 , . . . , b n be as in the statement of the proposition; let d q denote the degree of b q , and let b q denote its image in B dq ∼ = F 0 H dq (C)/F 1 H dq (C). The argument that the b q generate H(C) as an A-module is similar to the proof of (1), and we omit the details. To show that the b q are linearly independent over A, suppose we have a relation q a q b q = 0 with not all a q = 0. We may assume that each term a q b q has the same degree, which we denote n; then the degree of a q is c q := n−d q . Let c denote the smallest of the c q for which a q = 0. Then each term a q b q lies in F c H n (C). By definition of the product on E ∞ , the image of q a q b q in E c,n−c Since the multiplication map A⊗B → E ∞ is an isomorphism, this implies cq=c a q ⊗b q is zero in A ⊗ B. As the b q are linearly independent, this implies that all the a q with c q = c must be 0, contradicting our choice of c. We conclude that the b q are linearly independent, as desired.
6. The relative Hochschild-Serre spectral sequence The following theorem shows the existence of the Hochschild-Serre spectral sequence and identifies the edge maps.
Theorem 6.1. Let g be a Lie algebra. Let k be a subalgebra of g, reductive in g, and let I be an ideal of g. Let I k = I ∩ k. Let M be a g-module.
(1) There is a spectral sequence converging to H p+q (g, k; M ), and an isomorphism ψ: E pq 2 ψ → H p (g/I, k/I k ; H q (I, I k ; M )).
(2) The edge morphism E p0 2 → H p (g, k; M ) corresponds under the isomorphism ψ to the composition where the first map is the pullback induced by the projection (g, k) → (g/I, k/I k ), and the second map is induced by the g-module map M I → M .
(3) The edge morphism H q (g, k; M ) → E 0q 2 corresponds under the isomorphism ψ to the pullback i * : H q (g, k; M ) → H q (I, I k ; M ) g/I .

Part
(1) of this theorem is proved in Section 6.1. Part (2) is Proposition 6.6, and part (3) is Proposition 6.7. In Section 6.4 we will show that the spectral sequence is compatible with pairings of representations.
6.1. Some notation. We follow the notational conventions of Definition 4.5. We have The composition is s p , and by Lemma 3.7 it induces the isomorphism s p . Let s denote ⊕ p s p .
6.2. The differential d 0 . We begin by calculating the differential d 0 .
Proposition 6.2. Under the isomorphism s, the differential d 0 : Proof. Let e ∈ E pq 0 , and chooseê ∈ F p C p+q satisfying π pê = e. By definition, d 0 e = π p de, so s p d 0 e = s p dê. As s p is the restriction of S p , we apply Lemma 4.3 and find Asê ∈ F p C p+q , we have S p−1ê = 0 by Lemma 3.7. Therefore, proving the proposition.
The following diagram summarizes some of the relationships between the groups we are considering. The vertical arrows are inclusions, and the horizontal maps α and β (defined by this diagram) are the surjections from cocycles to cohomology.
. As a consequence of Proposition 6.2, we have: where the first equality is by Proposition 6.2, and the second by Proposition 5.4. A similar argument works to show s −1 p (C p (B q )) = B pq 1 .
Proof. Let e ∈ E pq 1 , and let φ(e) = α(z), with z ∈ C p (Z q ). Let z ∈ F p C p+q be the lift of z defined in Definition 3.12. Then s p ( z) = z, so by Corollary 6.3, z ∈ Z pq 1 . Observe that e = z + B pq 1 : this follows by Corollary 6.3, since φ( z + B pq 1 ) = α(s p z) = α(z) = φ(e). By Remark 5.5, d 1 e = d z + B p+1,q 1 , and therefore where the third equality is by Corollary 4.15, and the fourth is by definition of the map d + on cohomology. This proves the result.
where π p is surjective and φ is an isomorphism. By Proposition 6.4, φ : ker(d 1 ) → Z p (H q ) and φ : im(d 1 ) → B p (H q ) are isomorphisms. Thus, Proposition 5.4 implies that under φ • π p , the inverse image of Z p (H q ) is Z pq 2 and the inverse image of B p (H q ) is B pq 2 . The result follows.
6.3. Edge maps. In this section we show that the edge maps are compatible with maps defined using the functorial properties of Lie algebra cohomology.
Since There is a natural morphism where the first map is the pullback induced by the projection (g, k) → (g/I, k/I k ), and the second map is induced by the g-module map M I → M . More concretely, we can view C(g/I, k/I k ; M I ) as the subspace of elements f ∈ C(g, k; M ) such that f vanishes when any argument is in I, and such that the image of f lies in M I . We claim that F p Z p (g, k; M ) = Z p (g/I, k/I k ; M I ).
Since by definition of the filtration any element g of the left hand side vanishes when any argument is in I, to verify the claim we only need to check that any such element has image in M I . This follows because if x 0 ∈ I, then 0 = dg(x 0 , . . . , x p ) = x 0 g(x 1 , . . . , x p ).
Let f ∈ Z p (g/I, k/I k ; M I ). The map η takes the class of f in H p (g/I, k/I k ; M I ) to the class of the same element f , but now in H p (g, k; M ). We see that the image of η lies in F p H p (g, k; M ).
There is another morphism H p (H 0 ) → H p (g, k; M ) defined as the composition of ψ −1 with the edge morphism ǫ p : The next proposition shows that this morphism coincides with η.
Proposition 6.6. The following diagram commutes: Proof. As observed in equation (5.6), Let f ∈ F p Z p (g, k; M ) = Z p (g/I, k/I k ; M I ). Then ǫ p takes the class of f in E p0 2 to the class of f in H p (g, k; M ). On the other hand, s p f is just f , now viewed as an element of C p (g/I, k/I k ; C 0 (I, I k ; M )), and αs p f is again f , but now viewed as an element of Z p (g/I, k/I k ; M I ). Thus, ψ takes the class of f in E p0 2 to the class of f in H p (g/I, k/I k ; M I ). The discussion preceding the proposition shows that applying η to this yields the class of f in H p (g, k; M ). We conclude that the diagram commutes, as claimed.
We now consider the other edge morphism. By Proposition 2.11, the inclusion i : (I, I k ) → (g, k) induces a pullback i * : H q (g, k; M ) → H q (I, I k ; M ) g/I = H 0 (g/I; H q (I, I k ; M )) = H 0 (H q ).
On the other hand, we have a morphism defined as the composition The next proposition shows that these two morphisms agree.
Proposition 6.7. The following diagram commutes: H q (g, k; M ) ǫq / / i * compatibility with the differential implies that the cup product pairing descends to cohomology, yielding a pairing H p (g, k; M ) ⊗ H q (g, k; N ) → H p+q (g, k; P ).
Lemma 6.10. There is a pairing (the brackets denote the class of a cochain).
The pairing is compatible with the differential d r : Lemma 6.13. Let e ∈ E p 1 +q 1 r (M ) and e ′ ∈ E p 2 +q 2 r (N ). Then Proof. The differential d r is calculated by choosing representative cochains and applying the differential d. The lemma is then a consequence of (6.9).
There is a pairing H q 1 (I, I k ; M ) ⊗ H q 2 (I, I k ; N ) → H q 1 +q 2 (I, I k ; P ).
Since this is a g/I-module pairing, we obtain a corresponding pairing in (g/I, k/I k )cohomology (with notation as in (6.8)): ∪ → H p 1 +p 2 (H q 1 +q 2 (P )). (6.14) Note that this pairing is derived from a pairing on the level of cochains.
The next result relates this pairing with the spectral sequence pairing, extending Theorem 5 of [HS] to the relative setting.
Proposition 6.15. Let e ∈ E p 1 +q 1 r (M ) and e ′ ∈ E p 2 +q 2 r (N ). Then Proof. Keep the notation of the proof of Lemma 6.10. Let a ∈ F p 1 A p 1 +q 1 and b ∈ F p 2 B p 2 +q 2 represent e and e ′ , respectively. It suffices to prove that the pairing of cochains satisfies s p 1 (a) ∪ s p 2 (b) = (−1) p 2 q 1 s p 1 +p 2 (a ∪ b). This follows from the calculation in the proof of Theorem 3 of [HS]. 6.5. A tensor product decomposition. The natural pairing of the trivial representation F with itself induces a product on E 2 (F), making this space a bigraded ring. In this subsection, we describe a tensor product decomposition of E 2 (F), under a certain cohomological hypothesis. For simplicity, if the coefficient representation is the trivial representation F, we will generally omit it from the notation. Thus, for example, we write C n (g, k) and H n (g, k) for C n (g, k; F) and H n (g, k; F), respectively, and E 2 for E 2 (F).
Unless otherwise stated, we will assume H(g/I, k/I k ) ⊗ H(I, I k ) is given the product structure defined by where a 2 ∈ H p 2 (g/I, k/I k ) and b 1 ∈ H q 1 (I, I k ). We will denote the product defined by the same formula except without the factor of (−1) p 2 q 1 by * , and refer to this as the "unsigned product".
This map induces a cohomology isomorphism (also denoted f ) f : H p (g/I, k/I k ) ⊗ H q (I, I k ) g/I → H p (g/I, k/I k ; H q (I, I k ) g/I ).
The space H(g/I, k/I k ; H(I, I k ) g/I ) has a ring structure. In fact, the definition of the product (which is made using the cohomology pairing induced by a pairing of representations) is such that if we equip H(g/I, k/I k ) ⊗ H(I, I k ) g/I with the unsigned product, then f is an algebra isomorphism. The isomorphism of (6.18) also respects the algebra structure, so the composition is an algebra isomorphism.
Hence Ψ is compatible with products, so it is an algebra isomorphism.
Corollary 6.23. Keep the assumptions of Proposition 6.17.
(2) Suppose that the spectral sequence degenerates at E 2 . Then the pullback map π * : H(g/I, k/I k ) → H(g, k) is injective and the pullback map i * : H(g, k) → H * (I, I k ) g/I is surjective. The kernel of i * is the ideal J generated by elements of the form π * (a), where a is a positive degree homogeneous element of H(g/I, k/I k ).
Proof. Part (1) holds because the map (6.24) is the inverse of the map Ψ from Proposition 6.17. For part (2), the injectivity of π * and surjectivity of i * follow because these maps are identified with edge maps (cf. Propositions 6.6 and 6.7) and the corresponding edge maps have this property. The statement about the kernel of i * then follows from Proposition 5.8.

The Belkale-Kumar product
In this section we apply the Hochschild-Serre spectral sequence to prove Theorem 1.1, which shows that the cohomology of a generalized flag variety, equipped with the Belkale-Kumar product, has a structure analogous to the cohomology of a fiber bundle.
Throughout this section g is a complex semisimple Lie algebra. Let G denote the corresponding adjoint group, with B ⊃ H a Borel and maximal torus, respectively. Let P be a parabolic subgroup of G which contains B, and let m = dim H 2 (G/P ) (cohomology is taken with complex coefficients). For each t = (t 1 , . . . , t m ) ∈ C m , Belkale and Kumar construct a product ⊙ t on the space H * (G/P ), and show that it is closely related to the geometric Horn problem. In [EG], it was shown that the ring (H * (G/P ), ⊙ t ) is isomorphic to a certain relative Lie algebra cohomology ring.
To describe this ring we need some more notation. We denote the Lie algebra of an algebraic group by the corresponding fraktur letter. Let l be the Levi subalgebra of p containing h. Let W and W P denote the Weyl groups of g and l, respectively. Let {α 1 , . . . , α n } ⊂ h * be the simple roots corresponding to the positive system for which the roots of b are positive. Given M ⊂ {1, . . . , n}, let l M be the Levi subalgebra generated by h and the root spaces g ±α i for i ∈ M . Let u M,+ (resp. u M,− ) be the subalgebra spanned by the positive (resp. negative) root spaces not contained in l M . We assume that the simple roots are numbered so that l = l I for I = {α m+1 , . . . , α n }.
Given a subspace V of g, let V ∆ denote the image of V under the diagonal map g → g × g. Let u K = (u K,− × {0}) + ({0} × u K,+ ).
Define a subalgebra g K of g × g by g K = l K,∆ + u K .
Then u K is an ideal of g K , and g K / u K ∼ = l K . Moreover, g K contains l ∆ (since l K does).
For t = 0, we denote u K by u, and note that u = u − × {0} + {0} × u, where u is the nilradical of p, and u − is the opposite nilradical. Thus, for t = 0, g K = l ∆ + u, and H * (g K , l ∆ ; C) ∼ = H * ( u) L .
The following theorem describes the Belkale-Kumar product in terms of relative Lie algebra cohomology. The generic case and the case t = 0 are established in [BK]; the general case is in [EG].
We now apply the Hochschild-Serre spectral sequence for relative Lie algebra cohomology. By Theorem 6.1, there is a spectral sequence converging to H * (g K , l ∆ ) with E 2 -term H * (g K / u K , l ∆ ; H * ( u K , C)).

(7.2)
Note that in this spectral sequence, the role of I k is played by u K ∩ l ∆ = 0.
Proposition 7.3. The above spectral sequence degenerates at the E 2 -term.
Proof. By standard facts concerning spectral sequences, it suffices to prove that the dimension of the E 2 -term coincides with dim(H * (g K , l ∆ )), which by Corollary 3.18 of [EG] is |W P |. For this, we may identify g K / u K = l K,∆ . We decompose where the sum is over nontrivial irreducible representations of L K appearing in the space H * ( u K , C). For each V i , there is a Hochschild-Serre spectral sequence converging to H * (l K , l; V i ) with E 2 -term, H * (l K /z K , l/z K ; H * (z K , V i )). Since z K acts semisimply on V i , H * (z K , V i ) = 0 if z K acts nontrivially on V i . Hence the modules H * (z K , V i ) are a direct sum of nontrivial l K /z K -modules. Since l K /z K is semisimple, Theorem 28.1 from [CE] implies that H * (l K /z K , l/z K ; H * (z K , V i )) = 0. Hence the E 2 -term (7.2) coincides with H * (l K , l ∆ ; H * ( u K , C) L K ). (7.4) This last space coincides with H * (l K , l) ⊗ H * ( u K , C) L K .
It is well-known that dim H * (l K , l) = dim H * (P K /P ) = |W P K /W P |, where W P K is the Weyl group of L K (see [EG], Equation (3.10)). Also, by Theorem 5.14 from [Kos1], dim H * ( u K , C) L K = |W/W P K |. We conclude that the dimension of the E 2 term (7.2) is |W P K /W P | · |W/W P K | = |W P |, and the proposition follows.
(3) The cohomology ring H * (P K /P ) with the usual cup product is isomorphic to a graded subalgebra A of (H * (G/P ), ⊙ t ). Further, the ring (H * (G/P K ), ⊙ 0 ) ∼ = (H * (G/P ), ⊙ t )/I + , where I + is the ideal of (H * (G/P ), ⊙ t ) generated by positive degree elements of A.